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Spatially Explicit Markov Models for Designing Habitat Reserves

Spatially Explicit Markov Models for Designing Habitat Reserves. Michael Bevers, Curt H. Flather, Michael R. Taaffe, and Laurel E. Travis USDA Forest Service, USDA Forest Service, University of Minnesota, and Metropolitan State University Systems Analysis Forestry Symposium Chile 2002

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Spatially Explicit Markov Models for Designing Habitat Reserves

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  1. Spatially Explicit Markov Models for Designing Habitat Reserves Michael Bevers, Curt H. Flather, Michael R. Taaffe, and Laurel E. Travis USDA Forest Service, USDA Forest Service, University of Minnesota, and Metropolitan State University Systems Analysis Forestry Symposium Chile 2002 Punta de Tralca, Chile March 4 - 7, 2002

  2. Ferrets are born and migrate and die and stuff

  3. The Deterministic Differential Equation (DDE) Abundance Model

  4. DDE Model The MAPLE code (.mws) The MAPLE code (.pdf)

  5. DDE: Example I N P U T

  6. X1(t ) X2(t ) X3(t ) X4(t ) The Number of Ferrets in Territory i at time t

  7. Limitations of The DDE Model • Ferrets tendnot to beinfinitely divisible – at least not cheerfully. • Models are not(humanely) interpretable for small-populations.

  8. The DDE Assumptions

  9. The Stochastic Abundance Model- Infinite Capacity: Input

  10. The Stochastic Abundance Model- Infinite Capacity: Output

  11. The Kolmogorov forward equations

  12. The Kolmogorov forward equations

  13. Moment Diff ’l Equations (MDE’s)

  14. MDE ’s

  15. MDE ’s Closed set of MDE’s!

  16. Interpretable Not interpretable small populations MDE’s vs. DDE’s:

  17. The Stochastic Abundance Model - Infinite Capacity:Example (same as the DDE ) MAPLE 7 Code (.mws file) MAPLE 7 Code (.pdf file)

  18. I N P U T

  19. E[N1(t )] E[N2(t )] E[N3(t )] E[N4(t )] The Expected Number of Ferrets in Territory i at time t

  20. Not available in the DDE Model

  21. The Stochastic Abundance Model - Finite-Capacity: Input The same input as the infinite capacity model and

  22. The Stochastic Abundance Model - Finite-Capacity: Output

  23. The Kolmogorov forward equations

  24. The Kolmogorov forward equations Approximations needed for large-population, many-territory stochastic models!

  25. K Moment Diff ’l Equations (MDE ’s)

  26. K Moment Diff ’l Equations (MDE ’s) Not Closed !

  27. Closure Approximation

  28. K MDE ’s

  29. K MDE ’s Pseudo-Closed!

  30. The Stochastic Abundance Model - Finite Capacity: Examples MAPLE 7 Code (.mws file) MAPLE 7 Code (.pdf file)

  31. I N P U T

  32. E[N1(t )] E[N2(t )] E[N3(t )] The Expected Number of Ferrets in Territory i at time t

  33. Thank you for your attention!

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